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统计代写|统计推断代考Statistical Inference代写|STAT431 Mathematical probability

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统计代写|统计推断代考Statistical Inference代写|Ramsey

In order to develop a more rigorous framework for probability, we take a brief detour into an area of mathematics known as measure theory. The ideas here may seem a bit esoteric at first. Later we will see how they relate to our intuition about how probability should behave.

Measure
Consider a set, $\Psi$, and a subset, $A \subseteq \Psi$. We want to get some idea of the size of $A$. If $A$ is finite, one obvious way to do this is just to count the number of elements in $A$. Measures are functions acting on subsets that give us an idea of their size and generalise the notion of counting elements. Since a measure acts on subsets of the sample space, the domain for a measure will be a collection of subsets. In order to ensure that the measure can be defined sensibly, we need this collection to have certain properties.

统计代写|统计推断代考Statistical Inference代写|Probability measure

In this section we will show how the framework of section $2.2 .1$ allows us to develop a rigorous definition of probability. Measure gives us a sense of the size of a set. Probability tells us how likely an event is. We will put these two ideas together to define probability as a measure.

To define a measure we need a measurable space, that is, a set and a $\sigma$-algebra defined on the set. Our intuitive description of probability in section $2.1$ introduces the idea of a sample space, $\Omega$, the set of all possible outcomes of our experiment. We also define events as subsets of $\Omega$ containing outcomes that are of interest. From this setup we can generate a measurable space, $(\Omega, \mathcal{F})$, where $\mathcal{F}$ is a $\sigma$-algebra defined on $\Omega$. Here $\mathcal{F}$ is a collection of subsets of $\Omega$ (as usual), and we interpret the elements of $\mathcal{F}$ as being events. Thus, if $A \in \mathcal{F}$ then $A$ is an event. Remember that probability is always associated with events so $\mathcal{F}$ will be the domain for probability measure.
Definition 2.2.6 (Probability measure)
Given a measurable space $(\Omega, \mathcal{F})$, a probability measure on $(\Omega, \mathcal{F})$ is a measure $\mathrm{P}: \mathcal{F} \rightarrow[0,1]$ with the property that $\mathrm{P}(\Omega)=1$.

Note that, as we might expect, the definition restricts the codomain of $\mathrm{P}$ to be the unit interval, $[0,1]$. The triple consisting of a sample space, a collection of events (forming a $\sigma$-algebra on the sample space), and a probability measure, $(\Omega, \mathcal{F}, \mathrm{P})$, is referred to as a probability space.

We give two examples of functions which satisfy the conditions for probability measures. Showing that these functions are probability measures is part of Exercise $2.2$

统计推断代写

Measure

统计代写|统计推断代考统计推断代写|概率度量

. 在本节中，我们将展示$2.2 .1$节的框架如何允许我们对概率进行严格的定义。测量使我们对一组的大小有一种感觉。概率告诉我们一个事件发生的可能性。我们将把这两个概念结合起来，把概率定义为一种度量 要定义一个度量，我们需要一个可度量空间，即一个集合和一个在集合上定义的$\sigma$ -代数。我们在$2.1$部分对概率的直观描述引入了样本空间$\Omega$的概念，是我们实验的所有可能结果的集合。我们还将事件定义为$\Omega$的子集，其中包含感兴趣的结果。通过这个设置，我们可以生成一个可度量的空间$(\Omega, \mathcal{F})$，其中$\mathcal{F}$是在$\Omega$上定义的$\sigma$ -代数。这里$\mathcal{F}$是$\Omega$(和往常一样)的子集的集合，我们将$\mathcal{F}$的元素解释为事件。因此，如果$A \in \mathcal{F}$则$A$是一个事件。记住，概率总是与事件相关，因此$\mathcal{F}$将是概率度量的域。定义2.2.6(概率度量)

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。